Perfect Monotone Equilibrium

• Perfect monotone equilibrium is a solution concept in Bayesian games that requires players’ strategies to be monotone in type and robust to small perturbations.
• It employs trembling-hand completely mixed strategies, converging in the Prohorov metric to ensure asymptotically optimal and admissible responses.
• This equilibrium concept has practical applications in auction and pricing models, offering clear comparative statics and eliminating weakly dominated actions.

A perfect monotone equilibrium is a refinement of the monotone equilibrium concept for Bayesian games, distinguished by its robustness to small, unintended deviations (“trembles”) and the requirement that equilibrium strategies be monotone in player types and survive trembling-hand perturbations. This solution concept enhances both the credibility and admissibility of equilibrium predictions in auction, pricing, and resource allocation models where strategic complementarities and order structures are central features (He et al., 1 Sep 2025).

1. Definition and Motivation

A perfect monotone equilibrium is defined in Bayesian games with ordered type spaces (T₁, ..., Tₙ) and partially ordered or lattice-structured action spaces (A₁, ..., Aₙ). In a standard monotone equilibrium, a (pure) strategy profile is monotone if each player i’s mapping from type tᵢ to action aᵢ(tᵢ) is non-decreasing in the type order. However, this notion does not by itself eliminate equilibria supported by weakly dominated or inadmissible actions, which may be sensitive to perturbations in strategic behavior.

A pure strategy profile g is a perfect monotone equilibrium if it is the limit of a sequence {gᵏ} of completely mixed (i.e., trembling) strategies, with each gᵏ assigning positive probability to every open set of actions and converging to g (in the Prohorov metric on the space of probability measures). For each player and type, the sequence must nearly maximize interim payoffs:

• For any fixed tᵢ, limₖ→∞ ρ(gᵢᵏ(tᵢ), δ₍gᵢ(tᵢ)₎) = 0 (i.e., the mixed strategies concentrate at the pure strategy);
• The prescribed gᵢ(tᵢ) is an asymptotic best response to the trembling-hand strategies g₋ᵢᵏ.

This equilibrium concept ensures both monotonicity (players’ strategies are increasing in type) and trembling-hand perfection (robustness to small errors). In games with finite actions, such perfection guarantees that only admissible (undiominated) actions are played; for infinite actions, the notion extends to limit-admissibility.

2. Theoretical Framework and Conditions

The paper (He et al., 1 Sep 2025) establishes that the existence of a perfect monotone equilibrium is nontrivial and requires conditions stronger than those guaranteeing existence of a standard monotone equilibrium. The relevant technical setup includes:

• Ordered type spaces Tᵢ (e.g., intervals of ℝ), action spaces Aᵢ that are compact metric lattices or discrete ordered sets.
• Payoff functions uᵢ(a, t), bounded and continuous in actions.

Key conditions are:

• Increasing Differences Condition (IDC): For all actions aᵢ^H > aᵢ^L and types tᵢ^H > tᵢ^L,

$$[uᵢ(aᵢ^H, tᵢ^H; ... ) - uᵢ(aᵢ^L, tᵢ^H; ... )] \ge [uᵢ(aᵢ^H, tᵢ^L; ... ) - uᵢ(aᵢ^L, tᵢ^L; ... )].$$

This ensures that the incremental value of higher actions increases with type.

• Supermodularity: For multidimensional actions, supermodularity in own actions is required, not just quasi-supermodularity. This property enforces complementarity, eliminating weakly dominated equilibria and guaranteeing that best responses are monotone and robust to trembles.

The framework employs the Prohorov metric for weak convergence of measures, Helly’s selection theorem for compactness arguments, and best-response correspondence limits to construct perfect equilibria.

3. Limitations of Classical Conditions: Counterexamples

The inadequacy of classical conditions is demonstrated via two counterexamples:

• Single crossing condition (one-dimensional): While it ensures monotone best responses, it does not prevent equilibria supported by weakly dominated strategies. For example, in a first-price auction with binary actions, standard monotone equilibria may prescribe bidding profiles that include weakly dominated bids, which cannot be supported as trembling-hand perfect equilibria.
• Quasi-supermodularity (multidimensional): Even when combined with increasing differences, quasi-supermodularity fails to preclude limited equilibria reliant on non-robust strategies. The counterexample uses a player with a bid vector (over two items) facing a binary opponent, showing that certain equilibria persist under these weaker conditions, but collapse when supermodularity is required.

This demonstrates that only the combination of increasing differences and full supermodularity provides the monotonicity and robustness necessary for perfect monotone equilibrium existence.

4. Existence Theorem and Technical Formulation

The main existence theorem establishes that, under increasing differences and supermodularity (for the interim payoff with opponents playing monotone strategies), a perfect monotone equilibrium exists. The proof proceeds by:

• Constructing a sequence {gᵏ} of trembling-hand completely mixed monotone strategies converging in Prohorov metric,
• Verifying that these sequences yield asymptotic best responses, guaranteeing limit admissibility and monotonicity,
• Ensuring that in the limit, no weakly dominated (inadmissible) actions are played.

The limiting process ensures both monotone selection and trembling-hand perfection, aligning the measurable selection properties with the required equilibrium refinements.

5. Applications: Auctions and Pricing Models

The theoretical framework is applied to several canonical settings:

• Multiunit Uniform Price Auctions: Under the refined conditions, bidders’ equilibrium strategies are robust monotone mappings that eliminate weakly dominated bids. Bidder actions (bid vectors) are monotonic in types, and the equilibrium survives trembling-hand perturbations, resolving issues in existence proofs for auction models (cf. McAdams 2003).
• First-Price and All-Pay Auctions: In contrast to the classical single crossing-based existence theorems, only perfect monotone equilibria satisfy the admissibility and robustness criteria in the presence of type or payoff discontinuities.
• Bertrand Competition: Price-setting games with (unknown, possibly affiliated) costs exhibit monotone equilibrium strategies (prices increasing in cost), and under increasing differences plus supermodularity, these equilibria are perfect.
• Comparative Statics: The necessary conditions also guarantee monotone comparative statics, supporting economic analysis and policy design based on credible equilibrium selection.

Explicit construction of equilibrium profiles and comparative statics demonstrates the practical tractability of perfect monotone equilibria in standard market models.

6. Implications and Methodological Innovations

The introduction of perfect monotone equilibrium yields several key implications:

• Robust Refinement: The solution concept rules out equilibria resting on non-credible, weakly dominated actions and ensures predictions are stable under small deviations.
• Admissibility: In finite action settings, perfect monotone equilibria only support admissible strategies; in infinite (metric lattice) actions, the extension to limit-admissibility is used.
• Comparative Statics and Structural Prediction: The required stronger conditions for equilibrium existence naturally support comparative static results, maintaining monotonicity properties even under perturbations of economic primitives.
• Methodology: The use of the Prohorov metric for weak convergence, application of Helly’s selection theorem, and explicit treatment of perturbed strategies provide methodological templates for equilibrium construction in more complex settings such as discontinuous games or models with multidimensional strategies.

This equilibrium refinement closes the gap between existence theory and normative predictions in important economic environments, particularly those involving auctions and competitive pricing with strategic complementarities.

7. Conclusion

Perfect monotone equilibrium advances monotone equilibrium theory by imposing trembling-hand perfection, thereby ensuring robustness, undominated play, and meaningful monotonic comparative statics in Bayesian games with ordered structures. The existence theorem rests critically on the increasing differences and supermodularity properties of payoff functions, as weaker conditions may fail to guarantee robustness. The concept is concretely instantiated across a variety of auction and competition models, offering new foundations for equilibrium selection and analysis in economic settings where order and strategic complementarity are prevalent (He et al., 1 Sep 2025).